Three-dimensional X-ray imaging is based on taking several 1-D or 2-D projection images of a 3-D body from different directions. If 1-D projection images are available from all around a 2-D slice of the body with dense angular sampling, the inner structure of the slice can be determined. This is known as Computerized Tomography (CT) imaging technology, which is widely used in medical imagine today. A crucial part of CT technology is the reconstruction algorithm taking the X-ray images and producing a voxel representation of the 3-D body.
In the current commercial X-ray based 3D medical systems, such as CT, the attenuation of X-rays in one volume unit (voxel) inside the volume is typically defined basing on the values of the pixel values on projection images. This can be done, like in the known prior art, basing on the fact that each pixel value is an integral of attenuation along the X-ray beam in the volume. Therefore an inner structure and details of an object can be determined very accurately.
However, in many practical cases X-ray projection images are available only from a limited angle of view and therefore collecting sufficient information may be difficult. A collection of X-ray images of a 3-D body is called sparse projection data if (a) the images are taken from a limited angle of view or (b) there are only a small number of images. Sparse projection data does not contain sufficient information to completely describe the 3-D body.
However, some a priori information about the body is typically available without X-ray imaging, where the a priori information defines for example a shape or structure if the object to be imaged. A priori information may be achieved from previous measurements, for example, such as calculating an average of numerous measurements. Combining this information with sparse projection data enables more reliable 3-D reconstruction than is possible by using only the projection data.
There are, however, some problems when using traditional reconstruction algorithms such as filtered backprojection (FBP), Fourier reconstruction (FR) or algebraic reconstruction technique (ART), namely these do not give satisfactory reconstructions from sparse projection data. Reasons for this include requirement for dense full-angle sampling of data, difficulty to use a priori information for example non-negativity of the X-ray attenuation coefficient and poor robustness against measurement noise. For example the FBP method relies on summing up noise elements with fine sampling, leading to unnecessarily high radiation dose.
It is known a solution from the prior art (see WO 2004/019782, which is hereby incorporated into this document) for producing three-dimensional information of an object in medical X-ray imaging in which the object is modelled mathematically independently of X-ray imaging. The object is X-radiated from at least two different directions and the said X-radiation is detected to form projection data of the object. Said projection data and said mathematical modelling of the object are utilized in Bayesian inversion based on Bayes' formula
      p    ⁡          (              x        |        m            )        =                              p          pr                ⁡                  (          x          )                    ⁢              p        ⁡                  (                      m            |            x                    )                            p      ⁡              (        m        )            to produce three-dimensional information of the object, the prior distribution ppr(x) representing mathematical modelling of the object, x representing the object image vector, which comprises values of the X-ray attenuation coefficient inside the object, m representing projection data, the likelihood distribution p(m|x) representing the X-radiation attenuation model between the object image vector x and projection data m, p(m) being a normalization constant and the posteriori distribution p(x|m) representing the three-dimensional information of the object.
The solution described above is based on that biological issues have that kind of statistical a priori information that this information can be utilized successfully with Bayesian inversion in medical x-ray imaging. The suitable a priori information makes possible to model the biological tissue mathematically accurately enough and independently of X-ray imaging. From biological tissue it is possible to compile qualitative structural information, which makes it possible to utilize the Bayesian method successfully to solve the problems in medical three-dimensional x-ray imaging. There are certain regularities in biological tissues and this regularity is useful especially with the Bayesian method.
In generally one can say that iterative reconstruction methods are based on minimizing penalty functions, where two independent components are used, namely likelihood and a priory components, where the aim of the likelihood component is to minimize errors occurring when projection image determined from the reconstructed volume of the object is compared to the actual projection images of the same kind of object and volume, and where the aim of the a priory component is to minimize errors between the projection image determined from the reconstructed volume of the object and the mathematical model modelling the tissue of said volume of the object to be determined.
However, there are some problems with the prior art solutions, such as with the Bayesian method described above, namely they are not, as such, able to minimize or eliminate artifacts and especially artifacts occurring in parallel with the X-rays used in imaging process. Moreover, in some cases (such as in total variation) prior art Bayesian method even emphasizes or strengthens the artifacts in the direction of used X-rays, whereupon the quality of the reconstructed images is poor.